BLEED is the printing term that means that your art runs off the edge of your paper.
What does that mean?
When a design is first created, the designer makes a decision to either have the art end inside a "safe" zone, or continue all the way to the edge. Bleed comes into play when the art continues all the way to the edge. If the image ends even just a hair too soon, a white edge appears at the edge of the page (or card or whatever) and ruins the look of the image. So when the designer creates this masterpiece, she has to extend the art BEYOND where the printer will trim the piece off, eliminating any chance of a white strip showing. This extension is called the bleed.
Here is the difference between an image with bleed and the same without:
Do you HAVE to use a bleed?
No. You don't HAVE to. You can choose to have a colored image that ends before the edge and allow that white around it. If that is the case, that white becomes part of the composition, so it needs to work. In the examples below, I've shown two possible options. The left is where a white border is included as part of the composition. The right, I've eliminate a background altogether.
How to correctly set up a bleed.
First things first, check with your printer! They will have an amount that they recommend that works well for their setup and machinery. Typically it's a small amount, say 1/8" (0.125"), but can vary. There should also be a "safe zone" away from the cut. All the important stuff should stay within the safe zone and nothing super-duper important should go past it, just to, uh, be safe...